Run-length function of the Bolyai-Rényi expansion of real numbers

Abstract

By iterating the Bolyai-Rényi transformation T(x) = (x + 1)² (mod 1), almost every real number x ∈ [0, 1) can be expanded as a continued radical expression

$$x = - 1 + \sqrt {{x_1} + \sqrt {{x_2} + \ldots + \sqrt {{x_n} + \ldots}}}$$

with digits x_n ∈ {0, 1, 2} for all n ∈ ℕ. For any real number n ∈ [0, 1) and digit i ∈ {0, 1, 2}, let r_n(x, i) be the maximal length of consecutive i’s in the first n digits of the Bolyai-Rényi expansion of x. We study the asymptotic behavior of the run-length function r_n(x, i). We prove that for any digit i ∈ {0, 1, 2}, the Lebesgue measure of the set

$$D(i) = \left\{{x \in [0,1):\mathop {\lim}\limits_{n \to \infty} {{{r_n}(x,i)} \over {\log n}} = {1 \over {\log {\theta _i}}}} \right\}$$

is 1, where ${\theta _i} = 1 + \sqrt {4i + 1}$. We also obtain that the level set

$${E_\alpha}(i) = \left\{{x \in [0,1):\mathop {\lim}\limits_{n \to \infty} {{{r_n}(x,i)} \over {\log n}} = \alpha} \right\}$$

is of full Hausdorff dimension for any 0 ⩽ α ⩽ ∞.